William is 2 times as old as Ben. 35 years ago, William was 9 times as old as Ben. How old is William now?
Answer: We can use the given information to write down two equations that describe the ages of William and Ben. Let William's current age be $w$ and Ben's current age be $b$ The information in the first sentence can be expressed in the following equation: $w = 2b$ 35 years ago, William was $w - 35$ years old, and Ben was $b - 35$ years old. The information in the second sentence can be expressed in the following equation: $w - 35 = 9(b - 35)$ Now we have two independent equations, and we can solve for our two unknowns. Because we are looking for $w$ , it might be easiest to solve our first equation for $b$ and substitute it into our second equation. Solving our first equation for $b$ , we get: $b = w / 2$ . Substituting this into our second equation, we get: $w - 35 = 9($ $(w / 2)$ $- 35)$ which combines the information about $w$ from both of our original equations. Simplifying the right side of this equation, we get: $w - 35 = \dfrac{9}{2} w - 315$ Solving for $w$ , we get: $\dfrac{7}{2} w = 280$ $w = \dfrac{2}{7} \cdot 280 = 80$.